Download Entropy change of an ideal gas determination with no reversible

Revista Brasileira de Ensino de Fı́sica, v. 27, n. 2, p. 259 - 262, (2005)
www.sbfisica.org.br
Entropy change of an ideal gas determination with no reversible process
(Determinação da variação de entropia num gás ideal sem usar um processo reversı́vel)
Joaquim Anacleto1
Physics Department, University of Trás-os-Montes e Alto Douro, Vila Real, Portugal
Recebido em 26/10/2004; Aceito em 13/1/2005
As is stressed in literature [1], [2], the entropy change, ∆S, during a given irreversible process is determined
through the substitution of the actual process by a reversible one which carries the system between the same
equilibrium states. This can be done since entropy is a state function. However this may suggest to the students
the idea that this procedure is mandatory. We try to demystify this idea, showing that we can preserve the
original process. Another motivation for this paper is to emphasize the relevance of the reservoirs concept, in
particular the work reservoir, which is usually neglected in the literature2 . Starting by exploring briefly the symmetries associated to the first law of Thermodynamics, we obtain an equation which relates both the system and
neighborhood variables and allows entropy changes determination without using any auxiliary reversible process.
c which we believe to be
Then, simulations of an irreversible ideal gas process are presented using Mathematica°,
of pedagogical value in emphasizing the exposed ideas and clarifying some possible misunderstandings relating
to the difficult concept of entropy [4].
Keywords: First Law, symmetry, reversible process, entropy.
Como é referido na literatura [1], [2], a variação de entropia, ∆S, em um processo irreversı́vel é determinada substituindo o processo em causa por um outro, reversı́vel, que leve o sistema entre os mesmos estados
de equilı́brio. Isto pode ser feito pois a entropia é uma função de estado. Contudo, pode também sugerir aos
estudantes a idéia de que este procedimento é obrigatório. Tentamos desmistificar esta idéia, mostrando que
podemos preservar o processo original. Outra motivação para este artigo é sublinhar a relevância do conceito de
reservatório, em particular o de reservatório de trabalho, que é usualmente ignorado na literatura2 . Começando
por explorar brevemente as simetrias associadas à primeira lei da Termodinâmica, obtemos uma equação que
relaciona variáveis do sistema com variáveis da vizinhança e que nos permite determinar a variação de entropia
sem se recorrer a nenhum processo reversı́vel auxiliar. Seguidamente, considerando um gás ideal, apresentamos
c que cremos serem de valor pedagógico, porque
simulações de um processo irreversı́vel, usando o Mathematica°,
comprovam as idéias expostas e clarificam possı́veis questões relacionadas com o difı́cil conceito de entropia [4].
Palavras-chave: Primeira Lei, simetria, processo reversı́vel, entropia.
1. First Law as an interaction
dU = δ Q + δ W,
The first law relates the variation of the internal energy
of a system, ∆U , to the heat, Q, and work, W , flows
crossing its boundary. Adopting the point of view that
both heat and work are positive whenever these energies enter into the system3 , for an infinitesimal process
the first law may be written as [1]
1 E-mail:
(1)
where dU is an exact differential and δ Q and δ W are
inexact ones.
By considering a hydrostatic system4 and a reversible process, we can replace [1] δ Q by T dS and
δ W by −P dV , where T , S, P and V are the temperature, the entropy, the pressure and the volume, respectively. Therefore, Eq. (1) becomes
[email protected].
concept of heat reservoir is indeed required to set up the formalism of Thermodynamics, and this is sufficiently emphasized in
textbooks (see Refs. [1] and [2]). In contrast, the concept of work reservoir is not mentioned very often in the literature. However, an
important exception is [3].
3 Some authors [2] consider W as a positive quantity when energy leaves the system, following the historical development of Thermodynamics. See, for example, [5]. This paper includes a list of references which use different sign conventions for work.
4 Elementary work is generally expressed as the product of a generalized force (intensive variable) by differential of a generalized
displacement (extensive variable) (see Ref. [1]). However, by considering δW = −P dV we maintain the conceptual generality.
2 The
Copyright by the Sociedade Brasileira de Fı́sica. Printed in Brazil.
260
Anacleto
dU = T dS − P dV.
(2)
Analogously, the variation of the energy of the system neighborhood, dUn , is given by
dUn = Tn dSn − Pn dVn ,
(3)
5
where the subscript n means neighborhood . The energy conservation requires that dU = −dUn and Eqs.
(2) and (3) may be related as
T dS − P dV = −Tn dSn + Pn dVn .
P = Pn , and dV = −dVn ,
(5)
which could be called process symmetry, leads to the
conservation of the entropy, dS = −dSn .
Equations (5) imply that δ Q = T dS = −Tn dSn and
δ W = −P dV = Pn dVn , and the process is reversible.
The use of reservoirs is redundant, since no extra information is obtained from them, and the system variables
are quite sufficient to describe the process.
However, when conditions (5) are not simultaneously satisfied, not all terms in Eq. (4) express heat
or work exchanges. In this case the process is irreversible. Both the heat and work reservoir concept become not only useful but also essential to describe the
process, and accordingly all phenomena in the neighborhood can be considered reversible and therefore the
following equations remain valid,
δ Q = −Tn dSn ,
Tn
Pn
P
dSn +
dVn + dV,
(8)
T
T
T
which permits to obtain the entropy change for any
process, even irreversible, as we will see next.
dS = −
(4)
Even though preceding equation had been obtained
by imposing the restrictive condition of reversibility, it
will be valid also for an irreversible process, because all
quantities involved are state variables. This seems to
be a contradiction, and we believe, based on our teaching experience, that this issue is an important source of
difficulties.
An inspection of Eq. (4) shows its symmetric character: both sides are mathematically similar and expressed by the same thermodynamic quantities, and
their values depend only on the initial and final states.
This symmetry is related to the conservation of energy6 .
On the other hand, the symmetry expressed by
T = Tn ,
kinds of work exchanged: both the configuration and
the dissipative work. This is important to be pointed
out, as the work is expressed by thermodynamic variables of special systems – work reservoirs – that never
exhibit irreversible phenomena. Finally, from Eq. (4)
we have
2.
Simulation of a non-quasi-static
process of an ideal gas
Consider the following problem. An monatomic ideal
gas, initially at equilibrium with its neighborhood,
is characterized by the variables Vi = 0.01 m3 ,
Pi = 105 Pa, and Ti = 300 K. The gas is in a
diathermic container with a piston, as illustrated in
Fig. 1. Suddenly, the pressure and the temperature
of the neighborhood are changed to Pn = 4 × 105 Pa
and Tn = 200 K. Find the entropy change when the
final equilibrium state (Vf , Tf , Pf ) is reached.
The final equilibrium is attained when Tf = Tn
and Pf = Pn .
It is known [1] that the entropy change for a monatomic ideal gas is given by
∆S = 25 nR ln (Tf /Ti ) − nR ln (Pf /Pi ), where R is
the molar gas constant and n is the amount of substance. This formula, which was obtained by recurring
to a reversible process between the states (Ti , Pi ) and
(Tf , Pf ), gives ∆S = − 8.000 J K−1 .
The neighborhood £entropy change is given [1]¤
by ∆Sn = − (1/Tn ) 23 nR (Tf − Ti ) + Pn (Vf − Vi )
which gives ∆Sn = 19.168 J K−1 .
However, our goal is to obtain the entropy change,
both for the system and the neighborhood, without
recurring to any auxiliary reversible process. From
Eq. (8) and using the state equation P V = nRT we
obtain the equations we have to integrate numerically,
dS =
¢ nR
1¡
Pn dV + 23 nR dT +
T
V
(6)
dSn =
δ W = Pn dVn ,
(7)
in spite of δ Q 6= T dS and δ W 6= −P dV . This explains the contradiction mentioned formerly. Interesting enough, is the fact that the elementary work
δ W = Pn dVn does not depend on any variable of the
system, and assumes a unifying role since it includes all
5 Eqs.
µ
¶
Pn
1−
dV, (9)
P
1
[(P − Pn ) dV − T dS] .
Tn
(10)
The system attains the final state differently depending on the walls thermal conductivity, on the
strength of the friction at the piston, and on the relaxation efficiency of turbulent pressure gradients in the
gas7 . Nevertheless, the entropy changes are expected to
be exactly the same, irrespective to the way the system
(2) and (3) somehow already incorporate the second law due to the presence of the entropy.
conservation laws in Physics turn out to be closely connected to symmetries; see [6]. In field theory, for instance, Noether’s
theorem states that, for each invariance of the lagrangian, there is a conservation law, and vice-versa; see [7].
7 This consideration goes beyond the ideal gas model. See [8].
6 The
Entropy change of an ideal gas determination with no reversible process
reaches the final state, because those changes depend
only on the initial and final states and on the external
constrains.
Figure 1 - An ideal gas in an diathermic container with a piston.
By suddenly changing the initial neighborhood pressure and temperature, the system undertakes an irreversible process till a new
equilibrium state is attained.
To simulate the process, we consider a series of infinitesimal changes in the thermodynamic variables which
carry the system to the final state, as follows:
Let us consider the volume to be divided in NV
infinitesimal parts, each of which has the volume
dV = V /NV . Starting at V = Vi , a new value
Vnew = V ± dV is computed, taking the + sign if
Pn < P and the – sign if Pn > P . The temperature
variation has two contributions: dT1 = (Tn − T )/NT
(due to the heat exchange) and dT2 = −Pn dV /CV (due
to the work exchange), where CV = 32 nR and NT is a
261
integer like NV . So, starting at T = Ti , a new value
Tnew = T +dT1 +dT2 is computed. Finally, a new value
of the gas pressure is calculated using the state equation. Then, the new values are taken as the current
ones and the procedure is repeated till the conditions
|T − Tn | < 0.1 and |P − Pn | < 1.0 were satisfied as the
criterion for the final state reaching.
For each infinitesimal change on P , V and T we
compute the entropy changes, using Eqs. (9) and (10).
By adding up all them, from the initial to the final
state, we obtain ∆S and ∆Sn .
Integers NV and NT must be large enough to ensure
negligible computational errors. We can simulate an
isochoric process (δ W = 0) by considering NV → ∞,
and an adiabatic one (δ Q = 0) by considering NT → ∞.
Appropriate choices of NV and NT simulate different
processes, all irreversible, between the former limiting
cases.
Figures 2 and 3 show two simulations8 , labeled
c for
as (a) and (b), performed using Mathematica°
NV = 6000 and two different values of NT , NT = 2500,
Figure 2, and NT = 200, Figure 3. The graphs show
the time evolution of P , V , T and S, where the unit of
time corresponds to a calculation step.
c
Figure 2 - Simulation (a): NV = 6000, NT = 2500.
d
8 The
c notebook simulation program can be downloaded at the address http://home.utad.pt/∼anacleto/entropy.nb.
Mathematica°
262
Anacleto
Figure 3 - Simulation (b): NV = 6000, NT = 200.
d
In simulation (b) the gas reaches the final equilibrium much faster than in simulation (a). This happens
because in case (a) the walls have a lower thermal conductivity than in the case (b), which explains why the
temperature rises over 400 K, due to the strong gas
compression, before it decreases reaching the thermal
equilibrium at 200 K. As a consequence, the system entropy also rises before it decreases till its final value. In
case (b), the situation is reverse, the walls are good heat
conductors and the pressure starts decreasing, due to
the fast drop in temperature, and then it rises to its final equilibrium value. In both cases the overall entropy
changes are the same and agree with the values calculated before, within an error less than 0.05%. We can
reduce this error as much as we desire, by considering
extremely high values for NV and NT , but at an enormous computation time cost. Lots of simulations for
different values of NV and NT were performed and all
of them gave the same entropy changes, showing that
all processes considered are equivalent.
The simulations showed that entropy change can indeed be calculated without considering any auxiliary
reversible process, and they are also of didactical and
pedagogical value, since they can be used to follow the
evolution of the gas under a variety of different conditions by adequately choosing the variables values in the
program.
Acknowledgments
The author is indebted to Afonso Pinto and José Ferreira for valuable suggestions and for a critical reading
of the manuscript.
References
[1] M.W. Zemansky and R.H. Dittman, Heat and Thermodynamics (McGraw-Hill, Auckland, 1997), 7th edition.
[2] Enrico Fermi, Thermodynamics (Dover Publications,
Nova Iorque, 1937).
[3] H.B. Callen, Thermodynamics and an Introduction to
Thermostatics (John Wiley, New York, 1985), 2nd edition.
[4] D.F. Styer, Am. J. Phys. 68, 1090 (2000).
[5] Guy S.M. Moore, Phys. Education, 28, 228 (1993).
[6] R.P. Feynman, R.B. Leighton and M.L. Sands, The
Feynman Lectures on Physics (Addison-Wesley, Redwood City, 1963), v. I.
[7] C. Itzykson and J.B. Zuber, Quantum Field Theory
(McGraw Hill, Singapore, 1987).
[8] C.E. Mungan, Phys. Teach. 41, 450 (2003).